
An appropriate weak formulation then follows after substitution of the natural conditions bc2,trans2, and by searching for \(u(t,\mbox{{$x$}})\) , which, for almost every \(t\) , lies in the Sobolev space \(H^1_0(\Omega )\) , while its second time derivative is a member of the dual space \(H^{-1}(\Omega )\) . The space of test functions may also be widened to \(H^1_0(\Omega )\) , resulting in [1]}:
\(&\text{Find } u \in H^1\big (\mathcal {T};H^1_0(\Omega )\big ) \text{ s.t., for a.e. }t,\, \forall \, w \in H^1_0(\Omega ): \nonumber \\&\quad \big \langle \rho \frac{\partial ^2 }{\partial t^2}u , w \big \rangle _{H^{-1},H^{1}_0} + \int \limits _\Omega T\nabla u \cdot \nabla w \,\text{d}\Omega = - \int \limits _{\partial \Omega _N} \! g \, w \,\text{d}\Gamma \,, \) 
